Evolutionary Game Theoryand the Battle of the Sexes

Turid Bøe

WP 1997: 3

.

-IW orking Paper

Chr. Michelsen InstituteDevelopment Studies and Human Rights

Bergen Norway

ISSN 0804-3639

Evolutionary Game Theoryand the Battle of the Sexes

Turid Bøe

WP 1997: 3

Bergen, March 1997

Iiii ~~~~p~~~;d:~~~ ~~:2~~~E

-WorkingPaper WP -1997: 3

Evolutionary Game Theory and the Battle of the Sexes

Turid Bøe

Bergen, March 1997

Summary:This paper presents an outline of different approaches within evolutionary game theory and seesthese approaches in relation to the problem of choosing among multiple equilibria in normal formgares. More specifically the paper extends the analysis of Mailath, Kandori and Rob (1993)addressing 2x2 player symmetric gares to cover 2x2 player asymmetric garnes. Utilising anevolutionary model with a finite and equal number of players within two different playerpopulations and allowing for mutations perpetuating the system way from its deterministicevolution, we show that for the asymmetric "batte of the sexes" game the long run equilibrium (forlarge populations) chosen satisfies the Harsany and Selten (1988) criterion for risk dominance. Ingarnes with player-specific risk, that is in garnes where the players face identical risks regardless ofthe strategy chosen, but where the players from the different population face different risks, thepopulation size N does not influence the equilibrium chosen. In such garnes the long runequilibrium is the equilibrium with the largest surplus product.

Indexing terrns:Game theorySex rolesEconomic equilibriumDynamic processes

To be orderedfrom Chr. Michelsen Institute, Fantoftegen 38, N-5036 Fantoft, Bergen, Norway.Telephone: +4755574000. Telefax: +4755574166

ContentsIntroduction 1

Evolutionar game theory, an outlineThe static approachThe dynamic approachThe stochastic approach

223

6

Evolutionar game theoryand the battle of the sexes 9

Conc1uding remarks 16

Introductioni

Within traditional game theory "the battle of the sexes" (BoS from now on) is the name ofa game originating from Luce and Raiffa (1957). The gare models a situation in whichplayers wish to coordinate their behaviour, but have conflcting interests over outcome. Asimple two person, two strategy version of the game is captured by the following normalform representation.

2I Il1

I

Il2,1

0,00,01,2

Given that player 1 chooses strategy I the best player 2 can do is also to play strategy I andvice versa. The same is true for strategy Il. The strategy combinations (1,1) and (11,11) turn

out to be (strict) Nash equilibria for this strategic game2.

To find the equilibrium and to describe its characteristics in different settings, is in thecentre of economic analysis. Within gare theory, when modelling situations of strategicbehaviour, the concept of Nash equilbrium is seen as the one giving most insight (vanDamme; 1993).

There are growing awareness however, that the relevance of equilibrium analysis is not astraightforward one. The rationality assumption underlying traditional gare theoreticequilibrium concepts are under attach for being too unrealistic and restrictive. Questionslike "why is the use of Nash equilibrium appropriate" and "when wil such an equilbriumanalysis be suitable" are being raised and not easily answered. Traditional non-

cooperative game theory has also little to offer in situations where multiple equilbriaexist. In the exarple above, given the opportunity to choose, player 1 would prefer

equilibrium (1,1) to manifest itself, and player 2 equilibrium (11,11). But with regard to thepossibility of choosing among multiple equilibria, and eventually which one to choose,traditional game theory is silen t. As an example of a BoS gare, consider the formation ofhouseholds by men and women. In formng households wife and husband have a desire toco-operate, but the output obtained by the two pars may differ according to the type ofhousehold (eg. patriarchal, egalitarian) established3.

Ev01utionar game theory is a theoretical perspective inquiring whether players wilcoordinate their behaviour, if they wil play Nash and if so, which Nash equilibrium wil

1 The paper has benefited from the comments of Michihiro Kandori, Jørgen Weibull, Ugurhan Berkok andSjur Flåm.2 For a definition of a N ash equilibrium see. e.g. Osbome and Rubinstein (1994), p. 14.3 See e.g. Sen (1984) for a discussion of unequal distribution of goods within households.

1

be the outcome. This being the case, wil evolutionar game theory be able to predictwhich Nash equilibrium, if any, a society of agents playing the BoS game wil end up in?The paper proceeds as follows. First an outline of the different approaches withinevolutionar game theory is given4. Thereafter the insight gained by these differentapproaches wil be related to the BoS gare presented. The aim is to investigate how andto what degree the different evolutionar game theoretic approaches are ab1e to answer

the questions raised.

Evolutionary game theory, an outline.

Evolutionar game theory is originating, as the nare evolutionar implies, from biology

(Maynard Smith and Price; 1973). In its original formulation, the players have noconscious choice with regard to which strategy to play, but are programrned to playcertain strategies. These strategies may be modes of behaviour inherited from theirforbearers or assigned to them by mutation. Output is defined as reproductive capacity orfitness.

The sta tie approaeh.

The original formulation of the theory is a static approach which tries to capture a stableoutcome of an evolutionar process. Members of a single population N are randomlydrawn to interact with each other pairwise. In each match each player uses an actiondrawn from the set of available actions X. An output or fitness function u measures eachplayers ability to survive, usually assessed as its number of surviving offsprings. if theplayer uses action x when he faces the distribution d of actions of its potential opponents,then its ability to survive is measured by the expectation of u(x,z) under d, where x is theaction chosen by the player and z the possible actions chosen by its opponents. This

description corresponds to a two-player symmetric strategic garnes.

The concept of an evolutionar equilibrium is designed to capture the notion of a steadystate in which no mutant can invade the population. For every possible action x E X theevolutionar process occasionally transforms a small fraction of the population intomutants who follow y. In an equilibrium any such mutant must obtain an expected payofflower than that of the equilibrium action, so that the mutant dies out. If a fraction E :; O ofthe populationare mutants using the action y while all the other players use action x, then

4 For more comprehensive and techically sophisticated overviews, see Van Damme (1993), Mailath (1993),

Banerje and Weibull (1992), Hammersteinand Selten (1993) and Binmore and Samuelson (1993).5 A symmetric game has the following properties;

1) the number of pure strategies for each player is the same.

Sl = si = K = r1,2,...kl ß = ix E R+ \¿Xj = 11

2) the players position does not matter. U, (x,y)= Ui (y, x)

2

the average payoff to a non-mutant must exceed the average payoff to a mutant for allvalues of E sufficiently small;

u(x,(l- E)X + ey):; u(y,(l- E)X + ey) 6

An equilibrium strategy satisfying this condition, is denoted an evolutionarily stablestrategy (ESS).

Evolutionar game theory highlights that even in situations presuming no consciouschoice or rationality, there maybe a case for equilbrium analysis. .Further~-byemployingthe concept of ESS it may be possible to choose arong multiple Nash equilibria. In asymmetric game, only symmetric Nash equilibria are candidates for ESS? And among thesymmetric Nash equilibria, only the ones based on strategies satisfying the evolutionarstability condition are ESS.

In non-symmetric garnes however, ESS is of less use with respect to refinement of Nashequilibria. In gares with asymmetric player positions, only strict Nash equilibria can beESS (Selten; 1980)8. Garnes without strict Nash equilbrium fail to have ESS. Withregard to extensive form garnes, ESS is also of restricted value. An ESS has to reach allinformation sets in order to exc1ude alternative best responses (Selten; 1983).

The dynamie approaeh.

Within the dynaric approach the dynamics eventually leading to the equilibrium are ofinterest. The players, as they repeatedly interact, are assumed to try to maximise theirstage-game payoff. In doing so, their choice of strategy in the next stage game is based onthe belief that the distribution of their opponents play in the next game wil be the same asthe one revealed today. ff they tind it optimal, the players may choose to change strategybefore the next stage gare. In the general case, not all the players may adjust theirbehaviour in every period. Following Mailath (1993), given that the players, when theyadjust, adjust towards the best reply of last periods distribution, this periods fraction ofthe population (Pt) playing a given strategy x wil be given by the function;

Pi = b(pi_i)

where (p I-i) is the fraction of the population playing strategy x last period.

b(.) is referred to as the selection dynamics. By letting the selection dynamics satisfy theDarwinian property (that is; b(P) -: p when the expected output of playing strategy x is

6 With probability (l-e) a player encounter a non-mutant, and with probabilitye he encounters a mutant.7 A symmetr c Nash equilibrium is a Nash equilibrium where the strategy x is the best reply against itself.8 It z is the best response to strategy x, and no other strategy is the best response to x, then the equilibrium (x,z) is

a strict Nash equilibrium.

3

less than the expected output of an alternative strategy, and b(P) ~ p otherwise), andadding the conditions that b(O) = O, b(1) = 1 and the convention that b(P) = p when p issuch that the players are indifferentbetweenwhichstrategy to play (noadjustment in this

case), we get a first order difference equation describing the evolution of the fraction ofthe population playing strategy x9. The important feature of any selection dynamcs isthat it always adjust behaviour in the population towards the current best reply. Thepopulation dynamcs wil always, if possible, increase the fraction playing the best replyin the population.

The replicator dynamics (Taylor and Jonker; 1978) originating from evolutionar biologyis a paricular, continuous time, selection mechanism satisfying the criterion of alwaysadjusting population behaviour towards the current best reply. The dynamics for thepopulation shares Pi is given by;

Pi = (u(x¡ ,p) - u(p;p))p¡

where u(xi,p) is the expected payoff to any pure strategy i at a random match when thepopulation is in state p, andu(p,p) is the average payoff to an individual in the population

when the population is in state p10. The rate of change.! of the population share usingPi

strategy i equals the difference between the strategy's current payoff, and the currentaverage payoff in the population.

Within a biological context, the individual players are not assumed to adjust theirbehaviour. According to Weibull (1994), the replicators can be seen as the pure strategieswithin the gare. These strategies can be copied without error from parent to child. Insuch a setting the individuals in the interacting population are merely the hosts of thereplicators, and the replicator dynamics models how replicators compete for hosts in apopulation of pairwise interacting individuals. The replicator(s) resulting in the greatestbiological fitness or reproductive success wil be the winners. The population shareprogramred to a pure best reply to the current population state wil have the highestgrowth rate, and the sub-populations associated with better-than-average strategies wilgrow while strategies associated with worse-than-average strategies wil decline.

Compared to the static approach, dynamic evolutionar game theory shows not only thatthere may be a case for equilbrium analysis in economic analysis, but also how such astable situation may be the outcome of a dynamc process. Within the replicatordynamcs, Lyapunov stability deri ved without any rationality assumption, implies

9 Applied to the game gi ven in the introduction, with regards to strategy I played by group 1, b(p) o( p if p o( 113,

b(p) ~ p ifp ~ 113 and b(1I3) = 113.10 replicator dynamics in a multi-population setting is:

P¡h ='(U(Xih ,p_¡) - u¡(p))p¡

The growth rate for subpopulation husing strategy i.

4

aggregate behaviour which appear rational and coordinated in the sense of a symmetricNash equilbrium.

As in the static approach, the concept of ESS may be used to make refinements arongthe existing Nash equilibria. An ESS must be an asymptotically stable rest point of thesekind of dynamics. To be asymptotically stable is a stronger requirement than to beLyapunov stable 11. Not all rest points that are Lyapunov stable wil meet the requirementsof an asymptotically stable point. That is, not all Nash equilibria wil be ESS12.

However, the approach have serious limitations. The assumption that the players areprograrmed to play certain strategies may describe the behaviour of some simple organicorganisms (like bacteria), but hardly the behaviour of human beings. By letting theplayers be capable of adjusting their behaviour, the players get more human-likebehaviour. But this adjustment process is normally based on imitation of other playersstrategies, either randomly as in pure imitation or deliberately as in payoff dependentimitation. Explicit modellng of human learning processes are usually lacking, andbehaviour formation based on factors like role identification are not taken intoconsideration.

Evolutionar game theory reviewed thus far only tests the stability of the system againstsingle mutants. Mutants are assumed to be relatively rare, only one mutant comes intobeing at a time. The system wil thus have time to settle back into the original positionbefore the next mutant comes about. By assuming uniform random mixing of thepairwise players, the probability of meeting some specific type does not depend on yourown type. In a situation where mutants would mainly interact with each other, they mightenter the system more easily.

As in the static approach the usefulness of ESS is restricted outside the realm of onepopulation symmetric garnes. For asymmetric garnes the same critique as for the staticapproach applies, and within a multi-population setting there is no c1ear definition of ESS

(Weibull; 1994).

Neither the static nor the dynamc approach provides any solution to the problem of howto choose among strict Nash equilibria. Within the replicator dynamics any strict Nashequilibrium is an asymptotically stable stationar point of the dynamic (Weibull; 1993).Hence, the evolutionar process does not help in selecting among such equilibria. Thetheory cannot explain why beliefs are coordinated on a specific equilibrium. Having thiscoordination of beliefs as the point of deparure, the equilibrium chosen wil depend onthis departure point. The resulting equilbrium wil be path and history dependent.

11 For a discussion of Lyapunov and asymptotically stable rest points see Weibull (1994).12 Within the replicator dynamics, a Nash equilibrium is a stationary point of the dynamic. Each stable stationary

point is a Nash equilibrium, and an asymptotically stable fixed point is a pedect equilibrium (Bomze; 1986). It weallow for mixed strategies to be inherited, then asymptotically stable fixed points of the replicator dynamicscorrespond exactly to ESS (Bomze and Van Damme; 1982, Hines; 1980, Zeeman; 1981). It only pure strategi escan be inherited, being an ESS is sufficient but not necessary for asymptotic stability (Taylor and Jonker; 1978).

5

The stochastie approaeh.

Foster and Y oung (1990) and later Fudenberg and Haris (1992) and Kandori, Mailathand Rob (1993) showed that it may be possible to discriminate between strict Nashequilibria by adding perpetual randomness (or statistical noise) to the system. The modelsdiffer in their assumptions regarding time (continuous/discrete), population size

(finite/infinite) and dynamic formulation. We wil base the presentation of the stochasticapproach on the model of Kandori, Mailath and Rob (1993) (from now on KMR) whoassume discrete time and finite population size.

The gare is played by the members of a finite population of size N. In each period themembers of the population are randomly matched into pairs. Given two strategies, thestate of the system may be characterized by Zt, the number of agents playing a specificstrategy in a given time period. The set of possible values of Zt is Z = t O, 1,....,N l.

The players are capable of changing their strategies, but the opportunity to adjust thestrategy choice is assumed to arive stochastically and independently across players andtime. When the opportunity is there, the players are assumed to choose a myopic bestresponse, and a law of motion towards myopic best respons es is established. This law ofmotion, which satisfies the Darinian properties, is called the best response dynamic. In a2x2 gare, the dynamic is given by the rule:

~z) = r ;

if Jl ¡ (z) ~ Jl i (z)

if Jl¡(z)=Jli(z)

if Jl¡(Z) o: Jli(z)

where Jl ¡ (z) is the expected payoff by playing strategy I when the system is in state Z, and

Jl i (z) the expected output by playing strategy Il.

In addition to the best response dynamics, noise is added to the system. The noise comesabout by assuming that a player who is expected to play strategy x by the best responsedynamc, with a small but positive probabilty E mutates and plays strategy y13. Theprobabilty to mutate is assumed independent across players and over time.

13 In a setting with more than two strategy choices, the probability that a player expected to play strategy k mutates

to strategy j is mjE :; O, where e is the probability that the player wil mutate instead of playing according to the bestresponse dynamic and mj the probability that he will mutate to strategy j. I.jmj=l and e,m, j E (0,1)

6

The evolution of the system is now described by a Markov chain 14 which is finite,irreducible and aperiodic15. Given such a Markov chain all states have a strictly positiveprobabilty ofbeing observed in the limit as time tends to infinity.

In a situation without noise, the strict Nash equilibria of the gare wil be absorbing statesof the best reply dynamics. When noise is added, all states may be realized. All thesestates however, wil not have the sare likelihood of being observed. A Markov chainwhich is aperiodic and irreducible has a unique stationar distribution. This stationardistribution is stabile and ergodic. Stability implies that, staring at any initial distributionover states, the unconditional distribution of states converges to the stationar distributionasymptotically. Ergodicity on the other hand, means that the stationar distributionasymptotically describes the time average behaviour of the process. Given the propertiesof stability and ergodicity, when E is small, the probability of observing the system beingin one of the absorbing states is much bigger than the probability of observing it in anyother state. This being the case, the analysis of players coordinating their behaviour on aparicular equilibrium, may be confined to the set of stationar states only.

The goal of the stochastic analysis is to characterize the limit (stationar) distribution,given that it exists, when E goes to zero. States in the support of the limit distribution aredenoted long run equilbrium (KMR) or stochastically stable states (Foster and Y oung;1990).

Long run equilibrium (denoted LRE) is a stochastic equilibrium concept. This impliesthat the system, once settled down, wil not stay in the lon g run equilibrium for ever. Themutants make the system move from one equilibrium to another. The average time spentin the different equilibria however, wil differ, with the longest average time spent at thelong run equilbrium. The probabilty of observing the system in the long run equilibriumwil, as a result, be greater than the probabilty of observing the system in' any other state.

The long run equilibrium.

For general 2x2 symmetric garnes with two strict symmetric Nash equilibria, KMRshowed that the limit distribution places probabilty öne on the state in which all playersplay the strategy with the larger basin of attraction under the best reply dynamic. In otherwords, the equilbrium state with the biggest basin of attraction is the long runequilibrium.

To exemplify this result, consider the following 2x2 symmetric game played by 10players;

14 Within a Markov chain today's distribution of states depends only on yesterdays distribution.15 A Markov chain is irreducible if Prob( z(T)=z' I z(O)=z) )o O for all z and z' for som e T E N (N is the set of naturalnumbers). That is; any pair of states within the system are mutually reachable. It is aperiodie if the greatest commondivisor of tT E N I Prob( z(T)=z'l z(O)=z ))0 O L is L for all z and z'. There is no cycle in the dynamic.

7

2I Il1

I

Il2,2

0,0

0,01,1

The possible states Zt of this system (the number of agents playing strategy I) is 0,1,..,.10,and the steady states are O or 10. ff i' (the "mixed" strategy equilibrium) had been an

integer, i' would been the third steady state of the dynamic.

To find the basin of attraction belonging to the two equilbria, the critical level z* iscomputed. z* is the leve! for which the following is true (z* needs not be an integer):

sign (1l¡(Z) - 1l¡¡(z)) = sign (z - z*)

For general symmetric gares with payoff matrix:

2I Il1

I

Ila,a

c,b

b,c

d,d

N(d-b)+a -dKMR (1993) show ed that z* = . Essentially, z* corresponds to thea-c+d-bmixed strategy equilibrium which put probability J. = d - b on strategy i. z* isa-c+d-bnot exactly equal to ¡. however, since players with strategies I and Il face slightlydifferent strategy distributions due to the finiteness of the population. But as thepopulation size becomes larger, the difference between z* and ¡. vanishes.

The two states O and 10 have basins of attraction under b given by f z c: z* L and f z :; z* Lrespectively. The critical value z* divides the state space between 3 and 4. The basin ofattraction for equilbrium 10 consist of the states 4,..,10, and the basin of attraction forequilibrium O of the states 0,..,3. Given that the systemis in the equilibrium O (all playstrategy Il), at least 4 mutations are needed to reach the basin of attraction of equilibrium10. When staring in equilibrium 10 on the other hand, the minimum of 7 mutations arerequired to reach the other equilbrium16. It takes more mutations to upset equlibrium 10than to upset equilibrium O, resulting in 10 being the lon g run equilbrium.

i 6 In the general case, assuming z * = Jl, to escape the basin of attraetion of equilibrium (d,d) and reach the basin

of attraction of equilibrium (a,a), a number of dN/(a+d) mutations are needed. To go from equilibrium (a,a) to (d,d)on the other hand, N-(dN/(a+d))=aN/(a+d) mutations are required.

8

Given that the system is at one of the absorbing states, for the system to reach the basin ofattraction of the other absorbing state, it is crucial that enough mutations occur in oneperiod. Even though thereare othersequences ofeventsthat will take the system from

one absorbing state into the basin of attraction of the other, the lowest order probabiltyevent is the one in which the transition occurs in one period. To see this, assurne that themutations instead are spread over two periods. After the first mutation period and beforethe second, the system is stil in the basin of the original equilbrium. This being the case,at least one of the mutant players wil switch back to the original strategy by the bestresponse dynamic, and ths effect must be overcome by one more mutation.

Theorem 3 in KMR (1993;44) states that: given that the stage game is a coordinationgame and z*:f i', for any population size N;? 2 and any adjustment process satisfying

the Darwinian properties, the limit distribution puts probability 1 on N if z* -: i' and onO if z* :; i' .

In our exarple z* = LO~~~~~-i = 131 = 3l. Since N = 10 and z* -: i', the basin of

attraction supporting equilbrium N is bigger than the basin of attraction supporting

equilbrium O. All players choosing strateg y I is the long run equilbrium in this game.

Evolutionary game theory and the battle of the sexes

In this section we wil return to the questions raised in the introduction regarding the

abilty of evolutionar gare theory to predict which, if any, Nash equilibrium a BoS

gare, e.g. a game of men and women household builders, wil end up in.

The gare "the battle of the sexes" (BoS) put forward in the introduction, is anasymmetric, normal form game. The gare has two symmetric strict Nash equilbria inpure strategies, (1,1) and (I1,m, and a non-symmetric equilbrium in mixed strategies(X, X), (X, X) . In this asymmetric game, the two strict Nash equilbria are candidates forESS. By the use of a selection dynamics satisfying the Darwinian property, the law ofmotion in the gare is given by figure 1. The mixed strategy equilibrium turns out to beunstable, while the two strict Nash equilbria are asymptotically stable states of thesystem. In equilbrium A =(NxN) all players in both populations chooses to play strategy i.In A ' =(OxO) on the other hand, all of them chooses to play strategy Il.

9

X21

A1

A'1

XlI

Figure 1.

With respect to the game in question, the dynamic evolutionary approach predicts thesociety to end up in either A or A'. But which one, the theory cannot tell. To be able todetermne the most likely candidate, one has to move beyond the dynamcs of the modeland into the history of the game.

In 2x2 symmetric, . singe population garnes, the stochastic evolutionary approachformulated by KMR is capable of discriminating between different stationary states onthe basis of the number of states in support of each state 17. In the following we wil makeuse of their results in our inquiry to tind the lon g run equilbrium for the BoS gamepresented.

The BoS gare may be said to differ from the simple coordination game of the previoussection in two respects. First by being asymmetric rather than symmetric, and related tothis, by bein g played by players from two, rather than one, population. To capture thesedifferences, the technique employed to tind the basins of attraction corresponding to thedifferent stationar states in the 2x2 symmetric, one population game, must be modifiedto capture the characteristics of the game in question.

The modifications needed wil be presented in two steps. Westar with the ones requiredto go from a one population to a two population 2x2 symmetric game. Thereafter the

changes makng the technique suitable for asymmetric 2x2 garnes wil be put forward.

i 7 The cost of transition is measured by the number of mutations required to go from one equilibrium to the basin

of attraction of another in one leap. As N increases, the number of mutations needed also increases, making atransition less probable. This feature makes the analysis most suitable for relatively small populations. By intro-dueing the feature of local interaction among the players, the analysis is made independent of N (see.Ellison ; 1993)

10

Two populations, 2x2 symmetric game.

Imagine a 2x2symmetric game where the players playing against eachothercome fromtwo different populations, each containing N players. The state space for this gare

becomes Z = (0,1,.. . , N)x( 0,1,. .., N). The state space is a grid of points over a two

dimensional square. The possible states at each time is now described by a pair ofnumbers; Z = (ZI¡,Zii)' The state indicates the number of players in each population

playing strategy i (at time t).

For each population separately, the Darinian adjustment dynamics specified for the onepopulation setting, is assumed. Furthermore, since a player from population 1 is sure tomeet a player from population 2 within each stage gare, players belonging to population

1 are only interested in the distribution of strategies played by players of population 2,and vice versa. This being the case, a critical leve! z¡*, Z2* for each population can becomputed. The state space is now paritioned into four subregions. In subregions B and B'the dynamics are unambiguous. They point towards one of the game's two Nash

equilibria. In subregions C and D on the other hand, the Darinian . dynamics point inconflcting directions. The state space and the critical levels for a symmetric, twopopulation 2x2 gare is given in figure 2. The arrows indicate the Darinian dynamics.

Z21

NA

r- Lc B

ZZ*

;- j D ~A' Zii

Z¡* N

Figure 2.

To find the lon g run equilbrium, the cost of transition between the two equilibria A andA' must be computed. The co st of transition between the two equilibria is equal to theminimial number of mutations needed to move from one equilibrium into the basin ofattraction of the other. Under the assumption that the adjustment dynamics within the two

11

populations of players are identical, (implying that the speed of adjustment of population1 players are equal to speed of adjustment of population 2 players), the basin of attractionfor A and A' are given by are as B and B' (the areas with unambiguous dynamics)

respectively.

The cost of transition from A' to A, C(A',A) = Zi*18, where i = 1,219. To see this, supposethat all players initial ly choose strategy Il (equilibrium A J. ff suffcient number ofpopulation 1 players mutate towards strategy I, strategy I becomes the best reply forpopulation 2 players the next period. By the assumtion of stochastic adjustment, ithappens with positive probabilty that, in the next period, none of the players frompopulation 1 adjust but all players from population 2 switch to strategy i. In this way,equilbrium A, where I is played can be achieved by mutations in one of the populationsonly. By the same line ofreasoning, the cost oftransition from A toA', C(A,A') = N-z¡*. In

our example, z¡* is smaller than N-z¡*, and A becomes the long run equilibrium.

To ilustrate the effect of variable adjustment speed, consider the case where population 1players adjust intinitely faster than population 2 players in region D, and population 2players adjust infinitely faster than population i players in region C. In this case the basinof attraction for equilibrium A' consists of region B',C and D, while the basin of attractionfor equilibrium A consist of B alone. In this situation, to escape the A equilibrium, N-z¡*mutations are needed. To escape the A' equilibrium, region B must be reached. Thisrequires 2(z¡*) mutations (z¡* mutations in each population). To find the long run

equilibrium, N- z¡* must be compared to 2(z¡*). In theexarple presented, N- z¡* turns outto be smaller than 2( z¡ *), resulting in A i being the long run equilibrium.

Two populations, 2x2 asymmetric game.

In the asymmetric BoS gare, Zl * and Z2 * can be computed in the same way as in thesymmetri c two population 2x2 game. This time however, Zl* is not identical to Z2*. Thecritical leveIs, and the subregions of the state space exhibiting unambigious andconflcting dynamcs, is shown in figure 3. As before, given identical adjustment speedwithin the two populations, the areas B and B' exhibiting unambigous adjustment

dynamics are the basins of attraction supporting equilbrium A and A'.

18 This is an approximation. Since we are dealing with finite populations, the cost of transition is equal to the

integer just exeeding zi*' The same applies to N-zi*'19 In a symmetric two population game, zl * equa1s Z2*'

12

ZZI

NA

r iLc B

ZZ*

-i B' I:JA'

.

ZiiZ¡* N

Figure 3.

By the same line of reasoning as in the symmetric, two popu1ation case, the cost oftransition between the two equilibria are equal to;

C(A' ,A) = min(zi *,zi *) and C(A' ,A) = min(N - Zi *,N - Zi *).

ff min(zi*,zi *)-:min(N-Zi*,N-zi *), A wil be LRE and if min(zi*,zi *):;min( N - Zi * , N - zi *) A i is the long run equilibrium. In the example given,

min( Zi * , zi *) turn out to be equal to min( N - Zi *, N - zi *). The cost of transition from

A to A' is equal to the cost of transition from A i to A. This being the case, both the

equilbria are LRE. The system wil as aresult cyc1e between them, spending half the

time in each.

For general 2x2 symmetric gares, KMR showed that the limit distribution put probabilityone on the equilbrium with the biggest basin of attraction. We state that this result isapplicable also for 2x2 asymmetric garnes. For the BoS game in question, the basin ofattraction corresponding to equilibrium A i is of size (ZI *)( Zi *), and the bas in supporting

A is equal to (N - Zi *)( N - zi *). According to the size criteria, A is LRE if

(ZI *)(zi *) -: (N - Zi *)(N - zi *) and A' is LRE if the converse holds.

In contirmng this statement we wil only consider the case of(ZI *)(zi *) -: (N - Zi *)(N - zi *) since the case:; is just a re-Iabellng of the -: situation.We wil show that when (ZI *)(zi *) -: (N - Zi *)(N - zi *), the co st of transition in going

from A' to A is smaller than the co st of going from A to A'. Recall that the costs of

13

transition are given by; C(A',A)=min(ZI*,zi *) and C(A,A') = min(N-zi*,N-zi *).

For a transition to occur it is suffcient that enough mutations occur within one of thepopulations.

Proposition:

min(zpZ_I).( min(N - Zi )(N - Z_I) t: Zi + Z_I .( N t: ZIZ_I.( (N - Zi )(N - z_i)

Proor-

Observe that: Z¡ = min(zi 'Z_I) t: N - Z-i = min (N - Zi, N - Z_I)

Hence: Z¡ =min(zpz_i) .( N-z_i =min(N-ZpN-z_i)

t: Zi + Z_I .( N t: Pi + P-i .( 1( where p i = ~ )

t: PiP-i + Pi + P-i - PiP-i .( 1 t: PiP-i.( 1- Pi - P_i + PiP-i

t: PIP_i.((l-Pi)(l-p_l) t: zlz_I.((N-zi)(N-z_1)

The result shows that (ZI *)(Zi *).( (N - Zi *)(N - zi *) is equivalent tomin( Zi * , Zi *) .( min( N - Zi *, N - zi *) . The equilibrium supported by the bigger basin of

attraction is LRE. The result also shows that (ZI *)(zi *).( (N - Zi *)(N - zi *) isequivalent to z, * +zi * .( N. The criteria is a generalisation of the result obtained byKMR for 2x2 symmetri c garnes. In symmetric garnes, * and the criteria becomes

z* .( N/2.

The BoS gare given in the introduction is ilustrated in figure 4. For N = 10, Zi * +zi * =

10 = N. The basin of attraction corresponding to each equilibrium is of equal size. Boththe equilbria is LRE. The result is, as expected, equal to the result obtained by comparing

C(A',A) and C(A,A').

The res ult presented shows that when Zi * +zi * .( N, equilbrium A characterised by all the

players choosing strategy I wil be long run equilbrium. Generally, however, the sign ofZi * +zi * -N is a function of the risk characteristics of the BoS stage game. The riskcharacteristics is the risk connected with strategy I relative to strategy Il. ff c¡ is large relativeto bi, strategy I is riskier than strategy Il. Harsanyi and Selten (1988) proposed a notion of riskdominance to capture the risk characteristics of a 2x2 game; Their notion of risk dominatesstates that: Equilibrium A (all players play strategy l) risk dominates equilibrium A' (all

14

players play strategy Il) if (ai - c1 )( a2 - c2) :; (di - bi )( d2 - b2) and equilibrium A' risk

dominatesA if(ai -ci)(a2 -c2).;(dl -bl)(d2 -b2).

To obtain the risk characteristics of the BoS stage game, consider the game written ingeneral form;

2I IT

1I ai,az bi,cz

Il ci,bz di,d2

N(d. -b.)+a. -d.By makng use of the fact that Zj * = L L L L , where i = 1,2, it can be shown

aj -c¡ +dj -bjthat Zl * +Z2 * :: N if and only if:

(l-l-)(a -c )(a -c )-l-(a -b )(a -b ):;(l-l-)(d -b )(d -b )-l-(d -c )(d -c )N i i 22 NI 122- N i 122 NI 122As N increases, this condition approaches the Harsanyi & Selten notion of riskdominance.

For N big enough, that is for

(ai -bl)(a2 -b2)-(dl -ci)(d2 -c2)N;:l+(ai -ci)(a2 -c2)-(dl -bl)(d2 -b2)

the long run equilibrium chosen wil be the risk dominant equilbrium.

For gares with player-specific risks, that is for garnes where the players face identical riskregardless of the strategy chosen, but where players from the different populations facedifferent risk, bj = c¡, (i = 1,2), the condition fot Zi * +Z2 * :: N reduces to;

(ai -ci)(a2 -c2)::(d, -c,)(d2 -c2)

Sinc.e cj is the lowest output attainable, (kj - c¡ ), where k = a, d and i = 1,2, is the surplus

obtained given an equilbrium outcome. In a game with player-specific risk the long runequilbrium wil be the equilibrium with the largest surplus product. Note that for such garnesthe population size N do not influence the equilbrium chosen.

15

Should the risk be identical for the two groups of players, bj = cj = c, differences inequilbrium outputs determnes the long run equilibrium. This is most easily seen by choosingc= O. The condition thenreduces to aiai:; didi. The equilibrium with thelargest product of

outputs is the long run equilibrium.

Concluding Remarks

By extending the stochastic evolutionar model of KMR to cover the asymmetric 2x2BoS game, the lon g run equilibrium of the game is selected. For population sizessufficiently large the long run equilbrium is the risk dominant equilibrium. With respectto the specific game presented in the introduction, both the strict Nash equilbria turnedout to be LRE, indicating that the system wil cycle between them.

Within the model formulation presented, the size of the basin of attraction is theimportant feature of the selection dynamc. This feature however, is peculiar to discretemodels with independent mutations having equal probabilities to occur. The result maynot carr over to other model formulations.

Interesting aspects not captured by the model presented are:given differences in the mutation rates in different populations, which

relationship, if any, can be found between a fast/slow mutation rate and thecharacteristics of the long run equilibrium chosen.is the mutation rate exogenously given? What would happen if the tendency tomutate depends on the number of mutant players within the players ownpopulation group (the number of role models).wil the sizes of the interacting populations influence the equilibria chosen? In a

setting where the individuals are randomly drawn to interact pair-wise, unequalpopulation sizes must imply that not all individuals in the largest population getsto play in each stage gare. Wil the distribution of strategies within the subgroup

playing be representative for the group as a whole, or is there some selection

mechanism makng the distribution skewed toward a/some specific strategies?A game where the players only leam about the distribution of strategies played inthe opposite population, can model a situation where the game is assumed to takeplace only once in each generation. To catch the dynarics of a repeated gameplayed by the same two players in every period on the other hand, some learningprocess regarding the opponents strategy choices must be included.wil the assumption of players randomly mixed in pairs catch the team-formationprocedure in real life situations?

The list reveals many areas for research where the method of evolutionar game theorycan be applied and developed. Being able to predict not only the equilibrium statespossible, but also the one most probably chosen in different gare theoretic settings, is anexisting challenge increasing the usefulness of game theoretic analysis.

16

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Bomze, LM. and E. van Damme (1982): A Dynamical Characterization of Evolutionary Stable States.

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Economic Theory 57, 420-441.

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Harsanyi, J.c. and R. Selten (1988): A General Theory of Equilibrium Selection in Games, MIT Press.

Hines, W.G.S. (1980): Three Characterizations of Population Strategy Stability. Journal of Applied

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Kandori, M. and R. Rob (1994): Evolution of Equilbria in the Long Run: A General Theory andApplications. Mimeo Dept. of Economics, Princeton University and Dept. Of Economics,

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Luce, R.D. and H. Raiffa (1957): Games and Decisions. New York: John Wiley and Sons.

Mailath, G.J. (1993): Perpetual Randomness in Evolutionary Economics. CARESS Working Paper #93-01

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Osborne M.J. and A. Rubinstein (1994): A Course in Game Theory. The MIT Press.

Selten, R (1980): A Note on Evolutionary Stable Strategies in Asymmetric Animal Conflicts. Journal of

Theoretical Biology 84, 93-101.

Selten, R. (1983): Evolutionary Stability in Extensive 2-person Games. Mathematical Social Sciences 5,

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Taylor P.D. and L.B.Jonker, (1978): Evolutionarily Stable Strategies and Game Dynamics.Mathematical Bioscience 40, 145-156.

Van Damme, E. (1993): Evolutionary Game Theory. Center for Economic Research, Discussion Paper No.

9375, Tilburg Univ. The Netherlands.

Weibull, J.W. (1993): The "As if' approach to game theory: 3 positive results and 4 obstacles. Mimeo

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249-270.

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